How can I understand the bubble nucleation in the process of phase transition? Are there any analogs or clear figuration?

Thanks!

Well, a really easy to see example is boiling water. If you've ever boiled water, you may have noticed that bubbles start to form at certain spots on the container (these are generally imperfections in the surface), and as the bubbles progress towards the surface, they grow.

With boiling water, what happens is that the system as a whole would rather be in a gaseous state. But in order to transition to a gas, that transition has to start somewhere. With boiling water, it tends to start at some imperfection in the glass. And once the transition starts, it grows rapidly.

It's the exact same basic idea with phase transitions in the early universe. A phase transition first starts in some location, typically thought to be due to a quantum tunneling event, and then if that event leads to a lower-energy state, then the new phase will rapidly spread.